Vector Spaces

$\mathbf{R}^n$ and $\mathbf{C}^n$

Complex Numbers

We assume the root of $-1$ is $i$, ($i^2=-1$), which obeys the usual rules of arithmetic.

Definition

Complex Number

Definition

A complex number is an ordered pair $(a,b)$, where $a, b \in R$, but we will write this as $a+bi$.

The set of all complex numbers is denoted by $\mathbf{C}$:

$$\mathbf{C} = \{a + bi : a, b \in \mathbf{R} \}$$

Addition and Multiplication

Addition and multiplication on $\mathbf{C}$ are defined by

$$(a+bi)+(c+di)=(a+c)+(b+d)i,$$ $$(a+bi)(c+di)=(ac-bd)+(ad+bc)i;$$

here $a,b,c,d\in \mathbf{R}$.

Properties of Complex Arithmetic

Commutativity

$\alpha + \beta = \beta + \alpha$ and $\alpha\beta = \beta\alpha$ for all $\alpha, \beta \in \mathbf{C}$;

Associativity

$(\alpha + \beta) + \lambda = \alpha + (\beta + \lambda)$ and $(\alpha\beta) \lambda = \alpha (\beta\lambda)$ for all $\alpha, \beta, \lambda \in \mathbf{C}$;

Identities

$\lambda + 0 = \lambda$ and $\lambda 1 = \lambda$ for all $\lambda \in \mathbf{C}$;

Additive Inverse

For every $\alpha \in \mathbf{C}$, there exists a unique $\beta \in \mathbf{C}$ such that $\alpha + \beta = 0$;

Multiplicative Inverse

For every $\alpha \in \mathbf{C}$ with $\alpha \neq 0$, there exists a unique $\beta \in \mathbf{C}$ such that $\alpha\beta = 1$;

Distributive Property

$\lambda (\alpha + \beta) = \lambda\alpha + \lambda\beta$ for all $\alpha, \beta, \lambda \in \mathbf{C}$.

Let $\alpha, \beta \in \mathbf{C}$.

Let $-\alpha$ denote the additive inverse of $\alpha$. Thus $-\alpha$ is the unique complex number such that

$$\alpha + (-\alpha) = 0.$$

Subtraction on $\mathbf{C}$ is defined by

$$\beta - \alpha = \beta + (-\alpha).$$

For $\alpha \neq 0$, let $\dfrac{1}{\alpha}$ denote the multiplicative inverse of $\alpha$. Thus $\dfrac{1}{\alpha}$ is the unique complex number such that

$$\alpha(\dfrac{1}{\alpha}) = 1.$$

Division on $\mathbf{C}$ is defined by

$$\dfrac{\beta}{\alpha} = \beta (\dfrac{1}{\alpha}).$$

$\mathbf{F}^n$

info

In these documents, $\mathbf{F}$ stands for either $\mathbf{R}$ or $\mathbf{C}$. The letter $\mathbf{F}$ is used because $\mathbf{R}$ and $\mathbf{C}$ are examples of fields. Elements of $\mathbf{F}$ are called scalars, as opposed to a vectors.

Definition

For $\alpha \in \mathbf{F}$ and $m$ a positive integer, we define $\alpha^m$ to denote the product of $\alpha$ with itself $m$ times:

$$\alpha^m = \underbrace{\alpha \cdots \alpha}_{m \; \text{times}}.$$

Clearly $(\alpha^m)^n = \alpha^{mn}$ and $(\alpha\beta)^m = \alpha^m \beta^m$ for all $\alpha, \beta \in \mathbf{F}$ and all positive integers $m, n$.

Definition

$\mathbf{F}^n$ is the set of all lists of length $n$ of elements of $\mathbf{F}$:

$$\mathbf{F}^n = \{(x_1, \dots, x_2) : x_j \in \mathbf{F} \; \text{for} \; j = 1, \dots, n\}.$$

$x_j$ is the $j^{th}$ coordinate of $(x_1, \dots, x_n)$.

Addition and Scalar Multiplication

Addition in $\mathbf{F}$ is defined by adding corresponding coordinates:

$$(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n).$$

Scalar Multiplication The product of a number $\lambda$ and a vector in $\mathbf{F}^n$ is computed by multiplying each coordinate of the vector by $\lambda$:

$$\lambda (x_1, \dots, x_n) = (\lambda x_1, \dots, \lambda x_n)$$

here $\lambda \in \mathbf{F}$ and $(x_1,\dots,x_n) \in \mathbf{F}^n)$.

Properties of $\mathbf{F}^n$

Commutativity of Addition in $\mathbf{F}^n$ If $x, y \in \mathbf{F}^n$, then $x + y = y + x$.

Let $0$ denote the list of length $n$ whose coordinates are all $0$:

$$0 = (0, \dots, 0).$$

For $x \in \mathbf{F}^n$, the additive inverse of $x$, denoted $-x$, is the vector $-x \in \mathbf{F}^n$ such that

$$x + (-x) = 0.$$

In other words, if $x = (x_1, \dots, x_n)$, then $-x = (-x_1, \dots, -x_n$.

info

A field is a set of containing at least two distinct elements called 0 and 1, along with operations of addition and multiplication satisfying all the properties list here. Thus, $\mathbf{R}$ and $\mathbf{C}$ are fields, as is the set of rational numbers along with the usual operations of addition and multiplication. Another example of a field is the set $\{ 0, 1 \}$ with the usual operations of addition and multiplication except that $1 + 1 = 0$.

Vector Space

Definition

An addition on a set $V$ is a function that assigns an element $u + v \in V$ to each pair of elements $u, v \in V$.

A scalar multiplication on a set $V$ is a function that assigns an element $\lambda v \in V$ to each $\lambda \in \mathbf{F}$ and each $v \in V$.

Vector Space

Definition

A vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold:

Commutativity

$u + v = v + u$ for all $u, v \in V$;

Associativity

$(u + v) + w = u + (v + w)$ and $(ab)v = a(bv)$ for all $u, v, w \in V$ and all $a, b \in \mathbf{F}$;

Additive Identity

There exists an element $0 \in V$ such that $v + 0 = v$ for all $v \in V$;

Addictive Inverse

For every $v \in V$, there exists $w \in V$ such that $v + w = 0$;

Multiplicative Identity

$1v = v$ for all $v \in V$;

Distributive Properties

$a(u + v) = au + av$ and $(a + b) v = av + bv$ for all $a, b \in \mathbf{F}$ and all $u, v \in V$.

The simplest vector space is $\{ 0 \}$, which contains only one point.

$\mathbf{F}^\infty$ is defined to be the set of all sequences of elements of $\mathbf{F}$:

$$\mathbf{F}^\infty = \{ (x_1, x_2, \dots) : x_j \in \mathbf{F} \; \text{for} \; j = 1,2,\dots \}$$

Elements of a vector space are called vectors or points.

A vector space over $\mathbf{R}$, that is $\mathbf{R}^n$, is called real vector space; and a vector space over $\mathbf{C}$, that is $\mathbf{C}^n$, is called a complex vector space.

tip

If $S$ is a set, then $\mathbf{F}^S$ denotes the set of functions from $S$ to $\mathbf{F}$.

For $f, g \in \mathbf{F}^S$, the sum $f + g \in \mathbf{F}^S$ is the function defined by

$$(f + g)(x) = f(x) + g(x)$$

for all $x \in S$.

For $\lambda \in \mathbf{F}$ and $f \in \mathbf{F}^S$, the product $\lambda f \in \mathbf{F}^S$ is the function defined by

$$(\lambda f)(x) = \lambda f(x)$$

for all $x \in S$.